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I understand that torsion is a concept specific to three-dimensional spaces. Despite searching on Google, I've struggled to find how to extend the concept of torsion to an n-dimensional space.

Is it not feasible to adapt torsion for higher dimensions, or is there a formula in the academic literature that addresses this? I also looked for references on this topic but couldn't find any.

pedro colombino
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    It is hard to believe that a google search won't find torsion in differential geometry. – Kurt G. Feb 28 '24 at 11:26
  • @KurtG. Although I agree with you, I'm not sure exactly how torsion for affine connections is linked to the torsion of curves in $\Bbb R^3$, which is what I presume OP refers to – Didier Feb 28 '24 at 15:17
  • @KurtG. I think you misunderstood my post or maybe I wasn't clear. The torsion defined in a 3d space (7d too) can be found on google. I'm talking about generalizing the concept of torsion to any dimensional space (n-d). As a beginner in differential geometry, I may not have been able to identify this generalization in the literature. If you have references or libraries that have already developed this concept generalized to n-d space, please let me know. Thank you. – pedro colombino Feb 28 '24 at 16:51
  • This MO post discusses it. What is your understanding of torsion in 3D and 7D? – Kurt G. Feb 28 '24 at 17:12
  • Welcome to MSE! <> Torsion means multiple things in differential geometry. Can you please edit the question to clarify if you're asking about "torsion of a connection," "torsion of a space curve," or something else? – Andrew D. Hwang Feb 28 '24 at 17:17
  • @KurtG. In my case, I am interested in the torsion of multivariate curves. Here, torsion corresponds to the rotation speed of the binormal vector at a given point. This definition is true for 3D. For 7D, what I understood is that torsion is how much a curve deviates from lying in a 6-dimensional hyperplane at a given point. My view of torsion is the one explained here – pedro colombino Feb 29 '24 at 17:49
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    @AndrewD.Hwang I am rather interested in the torsion of curves in the context of the Frenet Frame – pedro colombino Feb 29 '24 at 17:52
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    If memory serves, there is a complete account in either volume 2 or (likelier?) 3 of Spivak's Comprehensive Introduction to Differential Geometry; the general idea is to define (an orthonormal generalization of) the Frenet frame recursively so that the derivative of the unit velocity $e_1$ lies in the plane spanned by ${e_1, e_2}$, the derivative of $e_2$ lies in the three-space spanned by ${e_1, e_2, e_3}$, and so forth. – Andrew D. Hwang Mar 01 '24 at 02:09
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    See this post and particularly this post. – Ted Shifrin Mar 05 '24 at 16:13

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